# arc length

*Arclength* is the of a section^{} of a differentiable^{} curve. Finding the length of an arc is useful in many applications, for the length of a curve can represent distance traveled, work, etc. It is commonly represented as $S$ or the differential $ds$ if one is differentiating or integrating with respect to change in arclength.

If one knows the vector function or parametric equations of a curve, finding the arclength is , as it can be given by the sum of the lengths of the tangent vectors^{} to the curve or

$${\int}_{a}^{b}|{\overrightarrow{F}}^{\prime}(t)|\mathit{d}t=S$$ |

Note that $t$ is an independent parameter. In Cartesian coordinates^{}, arclength can be calculated by the formula

$$S={\int}_{a}^{b}\sqrt{1+{({f}^{\prime}(x))}^{2}}\mathit{d}x$$ |

This formula is derived by viewing arclength as the Riemann sum

$$\underset{\mathrm{\Delta}x\to \mathrm{\infty}}{lim}\sum _{i=1}^{n}\sqrt{1+{f}^{\prime}{({x}_{i})}^{2}}\mathrm{\Delta}x$$ |

The term being summed is the length of an approximating secant to the curve over the distance $\mathrm{\Delta}x$. As $\mathrm{\Delta}x$ vanishes, the sum approaches the arclength, as desired. Arclength can also be derived for polar coordinates^{} from the general formula for vector functions given above. The result is

$$L={\int}_{a}^{b}\sqrt{r{(\theta )}^{2}+{({r}^{\prime}(\theta ))}^{2}}\mathit{d}\theta $$ |

Title | arc length^{} |

Canonical name | ArcLength |

Date of creation | 2013-03-22 12:02:43 |

Last modified on | 2013-03-22 12:02:43 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 14 |

Author | mathcam (2727) |

Entry type | Algorithm |

Classification | msc 26B15 |

Synonym | length of a curve |

Related topic | Rectifiable |

Related topic | IntegralRepresentationOfLengthOfSmoothCurve |

Related topic | StraightLineIsShortestCurveBetweenTwoPoints |

Related topic | PerimeterOfEllipse |

Related topic | Evolute2 |

Related topic | Cycloid^{} |